diff --git a/pandas/_libs/tslibs/timedeltas.pyx b/pandas/_libs/tslibs/timedeltas.pyx
index b5b3abd01328c..41925139e41ae 100644
--- a/pandas/_libs/tslibs/timedeltas.pyx
+++ b/pandas/_libs/tslibs/timedeltas.pyx
@@ -1213,6 +1213,12 @@ class Timedelta(_Timedelta):
                 return other.delta * self
             return NotImplemented
 
+        elif util.is_nan(other):
+            # i.e. np.nan, but also catch np.float64("NaN") which would
+            #  otherwise get caught by the hasattr(other, "dtype") branch
+            #  incorrectly return a np.timedelta64 object.
+            return NaT
+
         elif hasattr(other, 'dtype'):
             # ndarray-like
             return other * self.to_timedelta64()
@@ -1240,6 +1246,12 @@ class Timedelta(_Timedelta):
             # convert to Timedelta below
             pass
 
+        elif util.is_nan(other):
+            # i.e. np.nan, but also catch np.float64("NaN") which would
+            #  otherwise get caught by the hasattr(other, "dtype") branch
+            #  incorrectly return a np.timedelta64 object.
+            return NaT
+
         elif hasattr(other, 'dtype'):
             return self.to_timedelta64() / other
 
diff --git a/pandas/tests/scalar/timedelta/test_arithmetic.py b/pandas/tests/scalar/timedelta/test_arithmetic.py
index fce1ef29235cc..65709b0eebaf7 100644
--- a/pandas/tests/scalar/timedelta/test_arithmetic.py
+++ b/pandas/tests/scalar/timedelta/test_arithmetic.py
@@ -277,6 +277,14 @@ def test_td_mul_nat(self, op, td_nat):
         with pytest.raises(TypeError):
             op(td, td_nat)
 
+    @pytest.mark.parametrize('nan', [np.nan, np.float64('NaN'), float('nan')])
+    @pytest.mark.parametrize('op', [operator.mul, ops.rmul])
+    def test_td_mul_nan(self, op, nan):
+        # np.float64('NaN') has a 'dtype' attr, avoid treating as array
+        td = Timedelta(10, unit='d')
+        result = op(td, nan)
+        assert result is NaT
+
     @pytest.mark.parametrize('op', [operator.mul, ops.rmul])
     def test_td_mul_scalar(self, op):
         # GH#19738
@@ -328,6 +336,16 @@ def test_td_div_numeric_scalar(self):
         assert isinstance(result, Timedelta)
         assert result == Timedelta(days=2)
 
+    @pytest.mark.parametrize('nan', [np.nan, np.float64('NaN'), float('nan')])
+    def test_td_div_nan(self, nan):
+        # np.float64('NaN') has a 'dtype' attr, avoid treating as array
+        td = Timedelta(10, unit='d')
+        result = td / nan
+        assert result is NaT
+
+        result = td // nan
+        assert result is NaT
+
     # ---------------------------------------------------------------
     # Timedelta.__rdiv__